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Theorem pwex 3973
Description: Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
Hypothesis
Ref Expression
zfpowcl.1 𝐴 ∈ V
Assertion
Ref Expression
pwex 𝒫 𝐴 ∈ V

Proof of Theorem pwex
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 zfpowcl.1 . 2 𝐴 ∈ V
2 pweq 3403 . . 3 (𝑧 = 𝐴 → 𝒫 𝑧 = 𝒫 𝐴)
32eleq1d 2151 . 2 (𝑧 = 𝐴 → (𝒫 𝑧 ∈ V ↔ 𝒫 𝐴 ∈ V))
4 df-pw 3402 . . 3 𝒫 𝑧 = {𝑦𝑦𝑧}
5 axpow2 3970 . . . . . 6 𝑥𝑦(𝑦𝑧𝑦𝑥)
65bm1.3ii 3919 . . . . 5 𝑥𝑦(𝑦𝑥𝑦𝑧)
7 abeq2 2191 . . . . . 6 (𝑥 = {𝑦𝑦𝑧} ↔ ∀𝑦(𝑦𝑥𝑦𝑧))
87exbii 1537 . . . . 5 (∃𝑥 𝑥 = {𝑦𝑦𝑧} ↔ ∃𝑥𝑦(𝑦𝑥𝑦𝑧))
96, 8mpbir 144 . . . 4 𝑥 𝑥 = {𝑦𝑦𝑧}
109issetri 2617 . . 3 {𝑦𝑦𝑧} ∈ V
114, 10eqeltri 2155 . 2 𝒫 𝑧 ∈ V
121, 3, 11vtocl 2662 1 𝒫 𝐴 ∈ V
Colors of variables: wff set class
Syntax hints:  wb 103  wal 1283   = wceq 1285  wex 1422  wcel 1434  {cab 2069  Vcvv 2610  wss 2982  𝒫 cpw 3400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-v 2612  df-in 2988  df-ss 2995  df-pw 3402
This theorem is referenced by:  pwexg  3974  p0ex  3979  pp0ex  3980  ord3ex  3981  abexssex  5803  npex  6777  axcnex  7141  pnfxr  7285  mnfxr  7289  ixxex  9050
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